Measures of central tendency and dispersion

IB Mathematics: Analysis and Approaches SL· 4.4 Descriptive statistics· 40 min read

1. Measures of Central Tendency★★☆☆☆⏱ 15 min

Measures of central tendency describe the typical or center value of a data set. There are three main measures, each suited to different scenarios.

📘 Definition

Mean

$\overset{x}{ˉ}$ (sample), $μ$ (population)

The sum of all data values divided by the number of values. For grouped data, we use midpoints of class intervals to calculate an estimated mean.

📘 Definition

Median

The middle value of an ordered data set. For even $n$ , it is the average of the two middle values.

📘 Definition

Mode

The most frequently occurring value in a data set. Can be used for categorical data.

📐 Worked Example

Find the mean, median and mode for the ungrouped data set: $[2, 3, 5, 5, 7, 8, 10]$

1
Calculate the mean by summing values and dividing by $n = 7$ :
2
$\text{Mean} = \frac{2+3+5+5+7+8+10}{7} = \frac{40}{7} \approx 5.71$
3
Data is already ordered, $n = 7$ (odd) so the median is the $\frac{7 + 1}{2} = 4^{th}$ value:
4
Median = $5$
5
The most frequent value is $5$ , so the mode is $5$ .

📐 Worked Example

Calculate the estimated mean for this grouped frequency table: $0 < x \leq 10, f = 4$ ; $10 < x \leq 20, f = 8$ ; $20 < x \leq 30, f = 3$

1
Find the midpoint $x$ of each class interval: $5, 15, 25$
2
Calculate $\sum f x$ and $\sum f$ :
3
$\sum fx = (4 \times 5) + (8 \times 15) + (3 \times 25) = 20 + 120 + 75 = 215$
4
$\sum f = 4 + 8 + 3 = 15$
5
Calculate estimated mean:
6
$\bar{x} = \frac{215}{15} \approx 14.33$

Exam tip:

Always state that the mean for grouped data is an estimate, not an exact value, you may lose a mark if you forget this.

2. Measures of Dispersion★★★☆☆⏱ 20 min

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Measures of dispersion describe how spread out the data values are around the central tendency. Common measures include range, interquartile range (IQR), variance and standard deviation.

📘 Definition

Standard Deviation

The average distance of data values from the mean, measured in the same units as the original data. Variance is the square of standard deviation.

📐 Worked Example

Calculate the sample variance and standard deviation for the data set $[2, 4, 6]$

1
First calculate the sample mean:
2
$\bar{x} = \frac{2+4+6}{3} = 4$
3
Calculate the sum of squared deviations from the mean:
4
$\sum (x - \bar{x})^2 = (2-4)^2 + (4-4)^2 + (6-4)^2 = 4 + 0 + 4 = 8$
5
For sample variance, divide by $n - 1 = 2$ :
6
$s^2 = \frac{8}{2} = 4$
7
Sample standard deviation is the square root of variance:
8
$s = \sqrt{4} = 2$

3. Choosing Appropriate Measures★★★☆☆⏱ 15 min

The choice of measure depends on the shape of the distribution and presence of outliers. The mean is affected by extreme values, while the median and IQR are not.

📐 Worked Example

Hourly wages at a small store: $12, $12, $15, $16, $18, $22, $75 (manager's wage). Which measure of central tendency best represents the typical wage? Justify your answer.

1
Calculate all three measures: Mean ≈ $24.29, Median = $16, Mode = $12
2
Identify that $75 is an outlier, much higher than all other wages.
3
The outlier skews the mean, pulling it far above the wage of most employees. The mode is too low, and not representative.
4
Conclusion: The median is the best measure, as it is not affected by the extreme outlier and reflects the typical wage.

Use mean/standard deviation for symmetric, outlier-free data
Use median/IQR for skewed data or data with outliers
Use mode only for categorical data or discrete data with repeated values

Exam tip:

If a question asks you to justify your choice, you must explicitly mention outliers or skewness to get full marks.

4. Five Number Summary★★☆☆☆⏱ 10 min

The five number summary combines minimum, first quartile ( $Q_{1}$ ), median, third quartile ( $Q_{3}$ ) and maximum, used to construct box plots and identify outliers.

📘 Definition

Interquartile Range (IQR)

The spread of the middle 50% of the data, calculated as $I QR = Q_{3} - Q_{1}$

📐 Worked Example

Find the five number summary for test scores: $[45, 52, 60, 65, 68, 72, 75, 80, 85, 90]$

1
Minimum = 45, Maximum = 90, $n = 10$ even.
2
Median = average of 5th and 6th terms: $\frac{68 + 72}{2} = 70$
3
$Q_{1}$ = median of lower half (first 5 terms): 3rd term = 60
4
$Q_{3}$ = median of upper half (last 5 terms): 3rd term of upper half = 80
5
Five number summary: $[45, 60, 70, 80, 90]$ , $I QR = 20$

5. Common Pitfalls

Wrong move:

Reporting an exact mean for grouped data

Why:

Grouped data does not retain individual raw values, so midpoints are only an approximation

Correct move:

Explicitly state that your answer is an estimated mean for grouped data

Wrong move:

Confusing sample and population standard deviation from GDC output

Why:

IB examiners award full marks only for the value requested in the question

Correct move:

Always read the question carefully: use $s_{x}$ for sample, $σ_{x}$ for population

Wrong move:

Using mean for skewed data with outliers

Why:

Outliers pull the mean away from the center, making it unrepresentative

Correct move:

Use the median as the measure of central tendency for skewed data

Wrong move:

Forgetting to sort data before finding median or quartiles

Why:

Unordered data gives incorrect position values for median and quartiles

Correct move:

Always sort data from smallest to largest before calculating any positional measures

Wrong move:

Calculating IQR as $\frac{Q _{3} - Q _{1}}{2}$

Why:

IQR is the full difference between upper and lower quartile, not the half difference

Correct move:

Calculate IQR directly as $I QR = Q_{3} - Q_{1}$

6. Quick Reference Cheatsheet

Measure	Type	Key Notes
Mean (ungrouped)	Central Tendency	$\frac{\sum x}{n}$ , affected by outliers
Est. Mean (grouped)	Central Tendency	$\frac{\sum f x}{\sum f}$ , uses midpoints, estimate only
Median	Central Tendency	Middle ordered value, not affected by outliers
Mode	Central Tendency	Most frequent value, for categorical/discrete data
Range	Dispersion	$Max - Min$ , affected by outliers
IQR	Dispersion	$Q_{3} - Q_{1}$ , spread of middle 50%, robust to outliers
Sample Variance	Dispersion	$\frac{\sum ( x - x ˉ ) ^{2}}{n - 1}$ , for sample data
Population Variance	Dispersion	$\frac{\sum ( x - μ ) ^{2}}{N}$ , for full population

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Estimated mean for grouped data
2023 · 2
IQR and outlier identification
2021 · 1
Justify appropriate central tendency measure

What's Next

Measures of central tendency and dispersion are the foundation of all statistical analysis in IB AA SL. These concepts appear regularly in both Paper 1 and Paper 2, often combined with data representation, box plots, or cumulative frequency questions. You will use these skills to compare distributions, analyze bivariate relationships, and describe probability distributions in later topics. Mastery of these calculations and interpretation is essential for higher scoring statistics questions.